To solve for the height of the chest, we need to use the given information and the formula for the volume of a rectangular prism.
1. Given Information:
- Width ([tex]\(w\)[/tex]) of the chest: 16 inches
- Volume ([tex]\(V\)[/tex]) of the chest: 4096 cubic inches
- Length ([tex]\(l\)[/tex]) is 4 times the height ([tex]\(h\)[/tex])
2. Formulation:
- The volume [tex]\(V\)[/tex] of a rectangular prism is given by:
[tex]\[
V = l \times w \times h
\][/tex]
- According to the problem, the length [tex]\(l\)[/tex] is [tex]\(4 \times h\)[/tex]. Therefore, the equation becomes:
[tex]\[
V = (4h) \times 16 \times h
\][/tex]
- Substitute the given volume, 4096 cubic inches:
[tex]\[
4096 = 4h \times 16 \times h
\][/tex]
3. Simplification:
- Simplifying inside the equation:
[tex]\[
4096 = 64h^2
\][/tex]
- Divide both sides by 64 to isolate [tex]\(h^2\)[/tex]:
[tex]\[
h^2 = \frac{4096}{64}
\][/tex]
[tex]\[
h^2 = 64
\][/tex]
4. Solving for [tex]\(h\)[/tex]:
- Take the square root of both sides to solve for [tex]\(h\)[/tex]:
[tex]\[
h = \sqrt{64}
\][/tex]
- Considering both positive and negative solutions:
[tex]\[
h = \pm 8
\][/tex]
Given that a physical dimension like height must be a positive value, we discard the negative solution.
Therefore, the height of the chest is:
[tex]\[
\boxed{8}
\][/tex]
